How can I control UVs with a rig?
This might be helpful. UV Project where an empty can move and control the UV texture. You can either animate the empty directly, or parent it to another bone, or object, or...



p.s. if you press RR with the empty selected you can rotate the texture like an orbit.

Update: In response to extra blink question.

Above the initial shape I added a duplicate which floats just above the surface and repeated the process of adding a UV project - the difference is the second UV had a strip of alpha transparency.



My guess is more layers can be added for extra effect. Here I added some movement to the new UV alpha layer...



...and then added some scaling to origin UV eye texture.